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Mirrors > Home > MPE Home > Th. List > asclfval | Structured version Visualization version GIF version |
Description: Function value of the algebraic scalars function. (Contributed by Mario Carneiro, 8-Mar-2015.) |
Ref | Expression |
---|---|
asclfval.a | ⊢ 𝐴 = (algSc‘𝑊) |
asclfval.f | ⊢ 𝐹 = (Scalar‘𝑊) |
asclfval.k | ⊢ 𝐾 = (Base‘𝐹) |
asclfval.s | ⊢ · = ( ·𝑠 ‘𝑊) |
asclfval.o | ⊢ 1 = (1r‘𝑊) |
Ref | Expression |
---|---|
asclfval | ⊢ 𝐴 = (𝑥 ∈ 𝐾 ↦ (𝑥 · 1 )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | asclfval.a | . 2 ⊢ 𝐴 = (algSc‘𝑊) | |
2 | fveq2 6903 | . . . . . . . 8 ⊢ (𝑤 = 𝑊 → (Scalar‘𝑤) = (Scalar‘𝑊)) | |
3 | asclfval.f | . . . . . . . 8 ⊢ 𝐹 = (Scalar‘𝑊) | |
4 | 2, 3 | eqtr4di 2784 | . . . . . . 7 ⊢ (𝑤 = 𝑊 → (Scalar‘𝑤) = 𝐹) |
5 | 4 | fveq2d 6907 | . . . . . 6 ⊢ (𝑤 = 𝑊 → (Base‘(Scalar‘𝑤)) = (Base‘𝐹)) |
6 | asclfval.k | . . . . . 6 ⊢ 𝐾 = (Base‘𝐹) | |
7 | 5, 6 | eqtr4di 2784 | . . . . 5 ⊢ (𝑤 = 𝑊 → (Base‘(Scalar‘𝑤)) = 𝐾) |
8 | fveq2 6903 | . . . . . . 7 ⊢ (𝑤 = 𝑊 → ( ·𝑠 ‘𝑤) = ( ·𝑠 ‘𝑊)) | |
9 | asclfval.s | . . . . . . 7 ⊢ · = ( ·𝑠 ‘𝑊) | |
10 | 8, 9 | eqtr4di 2784 | . . . . . 6 ⊢ (𝑤 = 𝑊 → ( ·𝑠 ‘𝑤) = · ) |
11 | eqidd 2727 | . . . . . 6 ⊢ (𝑤 = 𝑊 → 𝑥 = 𝑥) | |
12 | fveq2 6903 | . . . . . . 7 ⊢ (𝑤 = 𝑊 → (1r‘𝑤) = (1r‘𝑊)) | |
13 | asclfval.o | . . . . . . 7 ⊢ 1 = (1r‘𝑊) | |
14 | 12, 13 | eqtr4di 2784 | . . . . . 6 ⊢ (𝑤 = 𝑊 → (1r‘𝑤) = 1 ) |
15 | 10, 11, 14 | oveq123d 7447 | . . . . 5 ⊢ (𝑤 = 𝑊 → (𝑥( ·𝑠 ‘𝑤)(1r‘𝑤)) = (𝑥 · 1 )) |
16 | 7, 15 | mpteq12dv 5246 | . . . 4 ⊢ (𝑤 = 𝑊 → (𝑥 ∈ (Base‘(Scalar‘𝑤)) ↦ (𝑥( ·𝑠 ‘𝑤)(1r‘𝑤))) = (𝑥 ∈ 𝐾 ↦ (𝑥 · 1 ))) |
17 | df-ascl 21855 | . . . 4 ⊢ algSc = (𝑤 ∈ V ↦ (𝑥 ∈ (Base‘(Scalar‘𝑤)) ↦ (𝑥( ·𝑠 ‘𝑤)(1r‘𝑤)))) | |
18 | 16, 17, 6 | mptfvmpt 7247 | . . 3 ⊢ (𝑊 ∈ V → (algSc‘𝑊) = (𝑥 ∈ 𝐾 ↦ (𝑥 · 1 ))) |
19 | fvprc 6895 | . . . . 5 ⊢ (¬ 𝑊 ∈ V → (algSc‘𝑊) = ∅) | |
20 | mpt0 6705 | . . . . 5 ⊢ (𝑥 ∈ ∅ ↦ (𝑥 · 1 )) = ∅ | |
21 | 19, 20 | eqtr4di 2784 | . . . 4 ⊢ (¬ 𝑊 ∈ V → (algSc‘𝑊) = (𝑥 ∈ ∅ ↦ (𝑥 · 1 ))) |
22 | fvprc 6895 | . . . . . . . . 9 ⊢ (¬ 𝑊 ∈ V → (Scalar‘𝑊) = ∅) | |
23 | 3, 22 | eqtrid 2778 | . . . . . . . 8 ⊢ (¬ 𝑊 ∈ V → 𝐹 = ∅) |
24 | 23 | fveq2d 6907 | . . . . . . 7 ⊢ (¬ 𝑊 ∈ V → (Base‘𝐹) = (Base‘∅)) |
25 | base0 17220 | . . . . . . 7 ⊢ ∅ = (Base‘∅) | |
26 | 24, 25 | eqtr4di 2784 | . . . . . 6 ⊢ (¬ 𝑊 ∈ V → (Base‘𝐹) = ∅) |
27 | 6, 26 | eqtrid 2778 | . . . . 5 ⊢ (¬ 𝑊 ∈ V → 𝐾 = ∅) |
28 | 27 | mpteq1d 5250 | . . . 4 ⊢ (¬ 𝑊 ∈ V → (𝑥 ∈ 𝐾 ↦ (𝑥 · 1 )) = (𝑥 ∈ ∅ ↦ (𝑥 · 1 ))) |
29 | 21, 28 | eqtr4d 2769 | . . 3 ⊢ (¬ 𝑊 ∈ V → (algSc‘𝑊) = (𝑥 ∈ 𝐾 ↦ (𝑥 · 1 ))) |
30 | 18, 29 | pm2.61i 182 | . 2 ⊢ (algSc‘𝑊) = (𝑥 ∈ 𝐾 ↦ (𝑥 · 1 )) |
31 | 1, 30 | eqtri 2754 | 1 ⊢ 𝐴 = (𝑥 ∈ 𝐾 ↦ (𝑥 · 1 )) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1534 ∈ wcel 2099 Vcvv 3462 ∅c0 4325 ↦ cmpt 5238 ‘cfv 6556 (class class class)co 7426 Basecbs 17215 Scalarcsca 17271 ·𝑠 cvsca 17272 1rcur 20166 algSccascl 21852 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5292 ax-sep 5306 ax-nul 5313 ax-pow 5371 ax-pr 5435 ax-un 7748 ax-cnex 11216 ax-1cn 11218 ax-addcl 11220 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4916 df-iun 5005 df-br 5156 df-opab 5218 df-mpt 5239 df-tr 5273 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5639 df-we 5641 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6314 df-ord 6381 df-on 6382 df-lim 6383 df-suc 6384 df-iota 6508 df-fun 6558 df-fn 6559 df-f 6560 df-f1 6561 df-fo 6562 df-f1o 6563 df-fv 6564 df-ov 7429 df-om 7879 df-2nd 8006 df-frecs 8298 df-wrecs 8329 df-recs 8403 df-rdg 8442 df-nn 12267 df-slot 17186 df-ndx 17198 df-base 17216 df-ascl 21855 |
This theorem is referenced by: asclval 21879 asclfn 21880 asclf 21881 rnascl 21890 ressascl 21895 asclpropd 21896 rnasclg 42169 |
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